A Fast Hierarchical Algorithm for 3-D Capacitance Extraction - CiteSeerX

A Fast Hierarchical Algorithm for 3-D Capacitance Extraction Weiping Shi, Jianguo Liuy, Naveen Kakani and Tiejun Yuz Dept. of Computer Science, Univ. of North Texas, Denton, TX 76203 y Dept. of Mathematics, Univ. of North Texas, Denton, TX 76203 z Dept. of Mathematics, Univ. of North Carolina, Charlotte, NC 28223

Abstract We present a new algorithm for computing the capacitance of three-dimensional perfect electrical conductors of complex structures. The new algorithm is signi cantly faster and uses much less memory than previous best algorithms, and is kernel independent. The new algorithm is based on a hierarchical algorithm for the n-body problem, and is an acceleration of the boundaryelement method for solving the integral equation associated with the capacitance extraction problem. The algorithm rst adaptively subdivides the conductor surfaces into panels according to an estimation of the potential coecients and a user-supplied error bound. The algorithm stores the potential coecient matrix in a hierarchical data structure of size O(n), although the matrix is size n2 if expanded explicitly, where n is the number of panels. The hierarchical data structure allows us to multiply the coecient matrix with any vector in O(n) time. Finally, we use a generalized minimal residual algorithm to solve m linear systems each of size n  n in O(mn) time, where m is the number of conductors. The new algorithm is implemented and the performance is compared with previous best algorithms. For the k  k bus example, our algorithm is 100 to 40 times faster than FastCap, and uses 1=100 to 1=60 of the memory used by FastCap. The results computed by the new algorithm are within 2.7% from that computed by FastCap. 1 Introduction In this paper, we study the capacitance extraction problem of three-dimensional perfect electrical conductors of complex structures. Fast and accurate capacitance estimation is important in the design of high-performance integrated circuits [8, 9, 10, 11, 12, 13, 15]. Two examples of three-dimensional complex structures for which capacitance strongly aects performance are DRAM cells and chip carriers used in high density packaging [10].

1.1 Integral Equation Approach We assume the conductors are embedded in a homogeneous dielectric medium, though our techniques can be extended to multiple dielectrics by using multi-layered Green functions. Since our algorithm is kernel independent, there is no need to introduce dielectric-dielectric interface conditions and additional variable, as required by kernel-dependent algorithms such as FastCap [11]. The capacitance of an m-conductor geometry can be summarized by an m  m capacitance matrix C. The diagonal entries Cii of C are positive, representing the selfcapacitance of conductor i, and the non-diagonal entries Cij are negative, representing the coupling capacitance between conductors i and j . To determine the j -th column of the capacitance matrix, we need only solve for the surface charges on each conductor produced by raising conductor j to unit potential while grounding rest of the conductors. Then Cij is numerically equal to the charge on conductor i. This procedure is repeated m times to compute all columns of C. Each of the m potential problems can be solved using an equivalent free-space formulation where the conductordielectric interfaces are replaced by a charge layer of density . Assuming a homogeneous dielectric, the charge layer in the free-space problem will be the induced charge in the original problem if  satis es the integral equation Z (x) = (x0) 4 kx1 , x0k da0 ; (1) 0 surfaces

where (x) is the known conductor surface 0potential, da0 is 3 the incremental conductor surface area, x, x 2 0< , and kx , x0 k is the Euclidean distance between x and x . The kernel of the integral equation is 1=kx , x0k. For simplicity, we will temporarily omit the scale factor 1=40 and reintroduce it later in Section 4. We use Galerkin scheme to numerically solve (1) for . In this approach, the conductor surfaces are divided into n small panels, and it is assumed that on each panel Ai , a charge qi is uniformly distributed. Then for each panel Ai , an equation is written which relates the known potential on Ai , denoted by vi , to the sum of the contribution of potential from charges on all n panels A1 ; : : : ; An . The result is a dense linear system Pq = v; (2) n n where q 2 < is the vector of panel charges, vn2n < is the vector of known panel potentials, and P 2 < is the

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35 Design Automation Conference ® Copyright ©1998 ACM 1-58113-049-x-98/0006/$3.50

DAC98 - 06/98 San Francisco, CA USA

potential coecient matrix where each entry Z Z 1 1 1 Pij = area(A ) daj dai ; i xi 2Ai area(Aj ) xj 2Aj kxi , xj k (3) for panels Ai and Aj . Matrix P is known to be positive, symmetric and positive de nite. The linear system of (2) has to be solved to compute panel charges, and the capacitances are derived by summing the panel charges. Direct methods based on triangularization of matrix P, such as Gaussian elimination and Cholesky factorization, require O2 (n3 ) operations. Iterative algorithms normally require O(n ) operations per iteration. These approaches are inecient if the number of panels is more than several thousands. In fact, any algorithm that uses the explicit representation of matrix P must use2 at least (n2 ) time and memory since the matrix is size n . Several algorithms have been proposed to solve (2) in sub-quadratic time. The FastCap algorithm of Nabors and White [10, 12] uses O(n) time, and is based on the O(n) time multipole algorithm for the n-body problem [6]. In this paper we propose a dierent O(n) time algorithm, which is based on a dierent O(n) time hierarchical algorithm for the n-body problem by Appel [1]. Ecient algorithms that are not based on the n-body problem include an FFT algorithm of Phillips and White [13], and the singular value decomposition (SVD) algorithm of Kapur and Zhao [9, 8]. The FFT algorithm and the SVD algorithm are about twice as fast as FastCap for test examples [13, 8], though the running time of both algorithms are O(n log n). 1.2 The n-Body Problem In the n-body problem, each of the n particles ,
exerts a force on all the other n , 1 particles, implying n2 pairwise interactions [1, 6]. The capacitance extraction problem shares many similarities with the n-body problem. Both the gravitational force and the electro-magnetic eld decreases as the distance increases, and the principle of superposition applies to both problems. The principle of superposition states that the potential due to a cluster of particles is the sum of the potential due to each individual particle. There are two types of algorithms for the n-body problem: the hierarchical algorithm of Appel [1] (1985) and the multipole algorithm of Greenberg and Rohklin [5] (1987). When Appel rst published his paper, he thought the time complexity was O(n log n). Later, a careful analysis shows the time complexity is O(n) [3]. At about the same time, Greengard and Rohklin [5] proposed the multipole algorithm, and proved its time complexity is O(n). The capacitance extraction algorithm FastCap is related to the multipole algorithm, while our algorithm is related to Appel's algorithm [1], and a radiosity algorithm [7]. Hanrahan, Salzman and Aupperle [7] used the ideas in Appel's algorithm to compute the re ection of light among a set of objects in computer graphics. Their experience shows Appel's algorithm is not only fast, but also very easy to implement. The hierarchical algorithm of Appel [1] uses the following two key ideas to compute all the forces on a particle: 1) For practical considerations, the forces acting on a particle need only be calculated to within the given precision. 2) The force due to a cluster of particles at some distance can be approximated, within the given precision, with a single term. However, there are several dierences between the capacitance extraction problem and the n-body problem, which prevent us from blindly adopting the n-body solution. One

major dierence is that in the n-body problem the objects are particles, while in the capacitance extraction problem, the objects are continuous conductor surfaces. Therefore, the hierarchical data structures are formed dierently. The n-body algorithm begins with n particles and clusters them into larger and larger groups. Our algorithm begins with a set of panels and subdivides the panels into smaller and smaller panels. (We can also start with a set of given small panels and build a hierarchical data structure using the same idea.) Another dierence is that the self-capacitance is very important in our problem while there is no such concept in the n-body problem. 1.3 Our Contribution In Section 2, we show how to build a hierarchical data structure for the potential coecient matrix by adaptively subdividing the conductor surfaces into subpanels according to the estimation of the coecient and a user-supplied error bound. This is dierent from the multipole algorithm where the conductor surfaces are divided in a pre-processing stage according to the geometry of each conductor individually. Our new algorithm subdivides the surfaces into panels of variable sizes, and it is guaranteed that all coecients are calculated to the same precision. More importantly, the coecient matrix contains O(n) block entries, where n is the number of panels. In Section 3, we show how to multiply the potential coecient matrix P with any vector in O(n) time, using the hierarchical data structure. The hierarchical data structure is of size O(n), but it implicitly stores the whole n  n potential coecient matrix. As a result, the multiplication can be done in O(n) time through three traversals of the hierarchical data structure. Then we use the Generalized Minimum Residual (GMRES) method to solve the linear systems. In Section 4, we present the experimental results that show the new algorithm is signi cantly faster than the multipole algorithm FastCap and uses signi cantly less memory, with a small dierence in the result compared with FastCap. 2 Potential Matrix Approximation This section describes the recursive re nement procedure which decomposes a large panel into a hierarchy of small panels, and builds a hierarchical representation of the potential matrix. We rst describe the procedure, then estimate the number of interactions that need to be considered, and nally analyze the error in the resulting potential matrix. Refine(Panel Ai, Panel Aj) { Pij = PotentialEstimate(Ai, Aj); Ri = longest side of Ai; Rj = longest side of Aj;

}

if ((Pij*Ri < Peps) AND (Pij*Rj < Peps)) RecordInteraction(Ai, Aj); else if (Ri > Rj) { Subdivide(Ai); Refine(Ai.left, Aj); Refine(Ai.right, Aj); } else { Subdivide(Aj); Refine(Ai, Aj.left); Refine(Ai, Aj.right); }

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Figure 2: Potential coecient matrix stored as links. Procedure PotentialEstimate returns an estimate of the potential factor from panel Ai to panel Aj de ned by (3). We use the closed form expressions derived by Wilton, et al [16] and numerical integration to compute the potential factor. If the estimated potential factor is less than the user provided error bound Peps (P ), then the panels are allowed to interact at this level of detail. The recursion is terminated and the interaction is recorded between the two panels. However, if the potential factor estimate is too large, then the estimate may not be accurate. In this case, the panel with the larger area, say Ai, is to be subdivided into Ai.left and Ai.right. The procedure Refine is called recursively. Procedure Subdivide subdivides a panel into two new panels. The subdivision hierarchy is stored in a binary tree with each node having two pointers left and right pointing to the two subpanels. Since a panel may be re ned against many panels, the actual subdivision of a panel may have occurred previously. When this happens, Subdivide does not perform any task. Figure 1 shows the re ne process of two conductor surfaces. We start with two conductor surfaces A and H in (a). Assume the coecient estimates between A and H are greater than the user provided error bound P , we subdivide A into B [ C , and then subdivide H into I [ J in (b). Now assume the estimates BJ; CI and CJ are less than P , but estimate BI is greater than P . Therefore, we record interactions BJ; CI and CJ at this level, while further subdivide panels B and I . The nal panels are shown in (e). We also compute the self-capacitance at this time. Note that if we use uniform grid partition, then there would be a total of 16 panels. Also note that we consider mainly the interaction with other panels, instead of the geometry.

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Figure 3: Potential coecient matrix with block entries. Figure 2 shows the hierarchical data structure produced by Refine, and associated potential coecients stored as links between the nodes in the hierarchy. Each binary tree represents one conductor surface, and each leaf node represents one nal panel that is not further divided in the discretization. The combination of all the leaf nodes completely cover all surfaces of the conductors. Each horizontal link represents one pair of potential coecients de ned in (3). Each self-link represents one self-potential coecient. Figure 3 shows the block matrix represented by the links of Figure 2. Each block entry represents one interaction between panels. Note that there are total 8 panels for the two conductors, so a traditional coecient matrix would require 64 entries. However, the block matrix has only 40 block entries, each stored as one oating-point number. We now show the block matrix always contains O(n) block entries, while the traditional matrix that explicitly lists all pairwise interactions must contain n2 entries. For simplicity, consider n panels of about the same size on the surface, and a binary tree constructed above the panels by merging adjacent panels recursively. Therefore, panels on the same level of the tree are of the same size. The error criterion in Refine says that two panels can interact directly only if Pij*R < Peps where R is the length of the longest side of the two panels. Since Pij is asymptotically 1=r where r is the distance between the two panels, the two panels can interact only if R=r < P , or r > R=P . For any xed P , this criterion is equivalent to the criterion that two panels at the same level in the tree can interact only if there are k other panels between them on that level, for some xed constant k. On the other hand, if panels Ai and Aj are too far, the ancestor of Ai would have interacted with the ancestor of Aj . Therefore, Ai and Aj cannot interact either. This argument applies to any level of the tree. Therefore, each node in the tree interacts with a constant number of other

nodes, and the total number of block entries is O(n). Esselink did a detailed analysis on the number of interactions by Appel's algorithm for the general case, and compared it with the multipole algorithm [3]. In Figure 2, it shows that each panel undergoes a constant number of interactions with other panels, regardless of their level in the tree. Large panels that are far apart interact directly, in the same way that small panels near each other interact directly. Finally, we analyze the relationship between the termination criteria and the accuracy of the computed potential factors. The termination criteria causes the potential coecient corresponding to each interaction to have approximately the same magnitude. This is because if an estimated potential factor were larger, the panels would be subdivided. More importantly, the termination criteria also places an upper bound on the error associated with the potential factor integral between any two interacting panels. Consider a panel A which contains two sub-panels A1 and A2 of similar shape and size. Let the radius of the smallest sphere that contains both A1 and A2 be R. Consider a point x0 of distance r from0 the center of the sphere, for some r > R. The potential at x due to the charge on panels A1 and A2 , with uniform charge densities 1 and 2 respectively, is Z Z 1 da1 + 2 da2 : (4) 0 0 x1 2A1 jjx , x1 jj x2 2A2 jjx , x2 jj If we treat A1 and A2 as a single panel A with 0 uniform charge density (1 + 2 )=2, then the potential at x will be: Z

1 + 2 1 da: (5) 0 x2A 2 jjx , xjj Assume without loss of generality 2  1 , then the dierence between (4) and (5) is Z

Z

2 , 1 1 1 0 , x1 jj da1 , 0 , x2 jj da2 2 jj x jj x A A 1 2  Z  1 1 2 , 1  , da1 0 0 2 AZ1 jjx , x1 jj jjx , x1 jj + R 1 da :  2 ,2 1 Rr 0 jj x , x1 jj 1 A1



Therefore the relative error is at most a constant times R=r. Since Pij is in proportion with 1=r, the condition Pij*R < Peps in procedure Refine implies that the error of the approximation for every entry is bounded by a constant times P : Therefore, as P goes to 0, the error goes to 0. Also, all entries in the coecient matrix are approximated to the same precision, which is a constant times P . In summary, our algorithm computes the potential factor matrix to within a xed error tolerance. In the process it organizes the potential factor matrix into blocks. The estimated potential factor associated with each block has the same error as other blocks. Furthermore, the total number of blocks grows linearly in the number of elements. 3 Solve Linear Systems 3.1 Fast Matrix-Vector Multiplication To solve the linear system Pq = v, an iterative algorithm requires the multiplication of the coecient matrix P with a

vector, which normally takes O(n2 ) time. However, because our P is represented by O(n) blocks, each matrix-vector multiplication can be done in O(n) time. The multiplication proceeds in three steps. To help understand the algorithm, please keep in mind P is the coecient matrix, q is the given charge vector, and the product Pq is the potential vector which we want to compute, though the algorithm works for any matrix and vector. In the following pseudo-codes, each panel A has two pointers left and right, pointing to the two sub-panels, and two elds charge and potential. For readers not familiar with recursive tree traversal, please see [2]. The rst step computes the charge for all interior nodes in the tree. The charge of an interior node is the sum of charges of its children panels. The charge of a leaf node Ai is given by Qi from q. This calculation can be done in a single depth- rst traversal of the tree propagating the charge upward. To compute the charge for each node, the charges of its children are computed rst, and then the charge of the node equals the sum of the charge of its children's. The time for computing the charge for all nodes is linear in terms of the number of nodes in the tree. AddCharge(Panel Ai) { if (Ai is leaf) Ai.charge = Qi; else { AddCharge(Ai.left); AddCharge(Ai.right); Ai.charge = Ai.left->charge + Ai.right->charge; } }

The second step computes for each panel Ai the potential due to its interacting panels. This can be computed by adding up the product of potential coecient Pij and charge at Aj , for all Aj that has interaction with Ai . The time for computing the charge for all nodes is linear in terms of the number of blocks in the coecient matrix. CollectPotential(Panel Ai) { for all Aj such that AiAj has interaction { Ai.potential = Ai.potential + Aj.charge*Pij; if (Ai is not leaf) { CollectPotential(Ai.left); CollectPotential(Ai.right); } } }

The third step distributes the potential from the interior nodes to the leaves. This is done by another depth rst traversal of the tree which propagates potential down to the leaf nodes. Each interior node adds its accumulated potential to its children's potential, recursively. The time of this step is linear in terms of the number of nodes in the tree. DistributePotential(Panel Ai) { if (Ai is not leaf) { Ai.left->potential = Ai.left->potential + Ai.potential; Ai.right->potential =

}

}

Ai.right->potential + Ai.potential; DistributePotential(Ai.left); DistributePotential(Ai.right);

The total time for the matrix-vector product is linear in terms of the number of nodes and links. It is well known that for any binary tree with n leaves, there are exactly n , 1 interior nodes. Therefore, the time is O(n), where n is the number of panels that are not further subdivided. 3.2 Generalized Minimum Residual Method We use the Generalized Minimum Residual (GMRES) method with restart [14] to solve the system of equations. The basic idea behind the GMRES method for solving an n  n linear system is to project the problem onto a Krylov subspace Kk of dimension k < n using the orthonormal basis constructed by a scheme due to Arnoldi [4], solve the k-dimensional subproblem using a standard approach, and then recover the solution of the original problem from the solution of the projected problem. The Arnoldi scheme involves the coecient matrix only multiplicatively. The dimension of the Krylov subspace is usually small. Since we can multiply P with any vector in O(n) time, each iteration can be computed in O(n) time. 4 Experimental Results The new algorithm is implemented and simulation results are reported in Tables 1 to 3. Both the new algorithm and the FastCap algorithm (for expansion order 0 and 2) are compiled and executed on a SUN Ultra SPARC Enterprise 4000. FastCap(0) is the fastest in the FastCap package, and is about twice as fast as FastCap(2). However FastCap(0) has 5% to 10% relative error with respect to FastCap(2). The test examples are k  k bus crossing conductors for k = 2 to 6, taken from the FastCap paper [10]. Each bus in k  k example is scaled to 1m  1m  (2k +1)m. The distance between buses on the same layer is 1m, and the distance between layers is 1m. The constant 40 is 111:27pF=m, according to [10]. Our GMRES reduces the two-norm of the residual to 1% of the initial residual, the same condition used by the conjugate residual algorithm in FastCap. The dierence or error of capacitance matrices is de ned as follows. Let the capacitance matrix computed by FastCap (2) be C and the capacitance matrix computed by another program be C0 . Then the dierence is estimated in the Frobenius norm [4]: jjC , C0 jj=jjCjj. This is the standard way to measure the dierence between two matrices. Table 1 compares the new algorithm with FastCap, and Table 2 and 3 show the rst row of the capacitance matrix computed by the new algorithm and by FastCap. Here is a summary of the comparison: 1) P = 0:01. Compared with FastCap(2), the new algorithm is 100 to 40 times faster, and uses 1=100 to 1=60 of the memory. The error is less than 2.7% with respect to FastCap(2). Compared with FastCap(0), our algorithm is 40 to 14 times faster, and is three times more accurate. (The memory usage for FastCap(0) appears to be incorrect.) 2) P = 0:003 and compared with FastCap(2). the new algorithm is 5 to 3 times faster, and uses

Table 1: Comparison for the bus problem. Time is seconds, memory is MB, and error is with respect to FastCap(2). Test Problems 2x2 3x3 4x4 5x5 6x6 FastCap (0) Time 4.3 11.0 41.3 39.0 84.4 Memory 12 16 86 24 97 Panel 792 1620 2736 4140 5832 Error 8.6% 5.0% 5.2% 8.5% 10.0 % FastCap (2) Time 10.1 22.9 51.1 161.8 Memory 5.6 16 26 46 Panel 792 1620 2736 4140

231.9 62 5832

New Algorithm (P = 0:01) Time 0.1 0.4 0.9 2.2 Memory 0.06 0.2 0.3 0.5 Panel 160 408 576 720 Error 2.1% 1.7% 1.8% 1.7%

5.8 0.9 1584 2.7%

New Algorithm (P = 0:003) Time 2.0 7.0 19.2 45.2 83.6 Memory 0.9 2.0 3.9 6.4 9.4 Panel 1888 3504 6720 10960 13152 Error 0.4% 0.2% 0.8% 0.6% 0.4% 1=6 to 1=8 of the memory. The error is less than 0.8% with respect to FastCap(2). We do not have access to the precorrected FFT algorithm [13], the MEI algorithm [15], and the SVD algorithm [8]. However, based on their relative performance to FastCap for the k  k bus problem, our algorithm is much faster than these algorithms as well. 5 Conclusion This paper presents a hierarchical algorithm that is signi cantly faster than previous best algorithms, and uses much less memory. The new algorithm is kernel independent, and therefore, is even more ecient when applied to multilayered dielectrics. The new algorithm does not require preprocessing to partition the conductor surface into panels, instead, it automatically partitions the panels according to a user supplied error bound. The new algorithm provides continuous tradeo of time with precision by changing the error bound P . The new algorithm is very simple, and provides many rooms for further improvements. There are several major dierences between our hierarchical algorithm and the multipole algorithm FastCap. 1) We partition the conductor surfaces according to an estimation of the coecients for the actual problem, while FastCap considers the geometry of each conductor separately. As a result, we use fewer panels than FastCap does, yet our precision is higher for the same expansion order. 2) To achieve high precision, we divide the conductor surfaces into smaller panels, while FastCap uses high order expansion terms. Experimental results show that it is less expensive to use more panels than to use more expansion terms. 3) We organize

Table 2: Comparison for 4x4 bus problem. Error is with respect to FastCap(2). FastCap l=0 l=2 New Algo P = 0:01 P = 0:003

C11

C12

First Row of Capacitance Matrix (pF) C13 C14 C15 C16 C17

C18

Error Time Memory (sec) (MB)

394.5 -124.0 -0.175 -2.471 -52.15 -43.39 -43.08 -52.92 405.2 -137.8 -11.91 -8.079 -48.36 -40.09 -40.01 -48.45

5.2%

41.3 51.1

86 26

401.6 -139.6 -9.180 -6.480 -48.46 -37.80 -37.53 -48.46 407.9 -135.7 -13.08 -8.720 -48.79 -41.20 -41.21 -48.72

1.8% 0.8%

0.9 19.2

0.3 3.9

Table 3: Comparison for 5x5 bus problem. Error is with respect to FastCap(2). FastCap l=0 l=2 New Algo P = 0:01 P = 0:003

C11

C12

C13

First Row of Capacitance Matrix (pF) C14 C15 C16 C17 C18

C19

C1;10

Error Time Mem (sec) (MB)

505.5 -142.8 -10.48 -20.31 2.21 -54.51 -50.13 -46.04 -50.32 -55.01 484.5 -166.1 -13.62 -6.17 -6.54 -48.84 -40.12 -40.12 -40.21 -48.90

8.5%

39.0 161.8

24 46

476.5 -168.2 -12.97 486.3 -165.5 -15.05

1.7% 0.6%

2.2 45.2

0.5 6.4

-1.91 -6.77

-4.48 -47.37 -38.52 -37.36 -38.41 -47.69 -6.01 -48.78 -40.49 -40.52 -40.52 -48.83

the coecient matrix more eciently using the hierarchical data structure. Some feathers, such as the adaptive evaluation de ned in FastCap [12], is free under our data structure. Acknowledgment The authors wish to thank Weishi Sun, Steve Tate and Dian Zhou for discussions and references, and anonymous referees for improving the presentation. References [1] A. A. Appel, \An ecient program for many-body simulation," SIAM J. Scienti c and Statistical Computing, Vol. 6, No. 1, 1985, 85{103. [2] T. H. Corman, C. E. Leiserson, and R. L. Rivest, Introduction to Algorithms, MIT Press, 1990. [3] K. Esselink, \The order of Appel's algorithm," Information Processing Letters, Vol. 41, 1992, 141{147. [4] G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, 1989. [5] L. Greengard and V. Rohklin, \A fast algorithm for particle simulations," J. Comp. Phys., Vol. 73, 1987, 325{348. [6] L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, Cambridge, Massachusetts, 1988. [7] P. Hanrahan, D. Salzman, L. Aupperle, \A rapid hierarchical radiosity algorithm," Computer Graphics (Proc. SIGGRAPH'91), Vol. 25, No. 4, July 1991, 197{206. [8] S. Kapur and D. E. Long, \IES 3 : A fast integral equation solver for ecient 3-dimensional extraction," Proc. 1997 ICCAD, 448{455.

[9] S. Kapur and J. Zhao, \A fast method of moments solver for ecient parameter extraction of MCMs," Proc. 34th DAC, June 1997, 141{146. [10] K. Nabors and J. White, \FastCap: A multipole accelerated 3-D capacitance extraction program," IEEE Trans. CAD, Vol. 10, No. 11, Nov. 1991, 1447{1459. [11] K. Nabors and J. White, \Multipole-accelerated 3-D capacitance extraction algorithms for structures with conformal dielectrics," Proc. 29th DAC, 1992, 710{715. [12] K. Nabors, et al., \Preconditioned, adaptive, multipoleaccelerated iterative methods for three-dimensional rst-kind integral equations of potential theory," SIAM J. Sci. Comput., Vol. 15, No. 3, May 1994, 713{735. [13] J. R. Phillips and J. White, \A precorrected FFT method for capacitance extraction of complicated 3-D structures," Proc. 1994 ICCAD, 268{271. [14] Y. Saad and M. H. Schultz, \GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., Vol. 7, No. 3, July 1986, 856{869. [15] W. Sun, W. W.-M. Dai, and W. Hong, \Fast parameters extraction of general 3-D interconnects using geometry independent measured equation of invariantce," Proc. 33rd DAC, 1996. [16] D. R. Wilton, et al., \Potential integration for uniform and linear source distributions on polygonal and polyhedral domains," IEEE Trans. Antennas and Propagation, Vol. AP-32, No. 3, March 1984, 276{281.